Suppose \(y\) is a function of \(x\), then set of all value of \(x\), for which \(y\) is defined, is known as domain of the function \(y = f(x)\). To calculate domain we must closely look on those values which can make \(y\), undefined, those values are not part of domain. For example if \(y = \frac{{x - 3}}{{x + 4}}\) clearly \(x + {\rm{ }}4 \ne 0\), have domain of the function is set of all real values except \(-4\).       

There are certain important points regarding domain of a composite function.

\({\rm{dom}}\left\{ {{\rm{ }}f\left( x \right){\rm{ }} \pm {\rm{ }}g\left( x \right){\rm{ }}} \right\}{\rm{ }} = {\rm{dom}}f(x) \cap {\rm{dom}}g(x)\) 

\({\rm{dom}}\left\{ {f\left( x \right).{\rm{ }}g(x)} \right\} = {\rm{dom}} f(x) \cap {\rm{dom}} g(x)\)

\({\rm{dom}}\left( {\frac{{f(x)}}{{g(x)}}} \right)\) \( = {\rm{dom}}f(x) \cap {\rm{dom}}g(x)\) and \(g\left( x \right) \ne 0\).

For example domain of  \(\log x + \frac{1}{{x - 1}}\) will be all values of \(x\), which are domain of \(\log x\) and \(\frac{1}{{x - 1}}\) both. Hence domain of \(\log x + \frac{1}{{x - 1}}\) is \(0 < x < 1\) and \(1{\rm{ }} < x < \infty \)